Neural-Like Dynamics in a Phase-Locked Loop System with Delayed Feedback
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-Like Dynamics in a Phase-Locked Loop System with Delayed Feedback I. V. Sysoeva,b*, M. V. Sysoevaa,c, V. I. Ponomarenkoa,b, and M. D. Prokhorova a
Kotelnikov Institute of Radio Engineering and Electronics (Saratov Branch), Russian Academy of Sciences, Saratov, 410019 Russia b Saratov State University, Saratov, 410012 Russia c Yuri Gagarin State Technical University of Saratov, Saratov, 410054 Russia *e-mail: [email protected] Received February 27, 2020; revised April 16, 2020; accepted April 16, 2020
Abstract—We proposed a model of neural dynamics based on a phase-locked loop system with delayed feedback. The model can demonstrate chaotic oscillation modes, in which switching between qualitatively different types of oscillations that are characteristic of neural activity takes place. Keywords: phase-locked loop system, delayed feedback, neural dynamics, intermittency. DOI: 10.1134/S1063785020070287
The task of constructing and researching mathematical models describing the dynamics of neurons has long attracted much attention [1]. The most famous dynamic models of neural activity are the Hodgkin–Huxley, FitzHugh–Nagumo, Morris– Lecar, and Hindmarsh–Rose models, described by ordinary differential equations, and the Izhikevich, Rulkov, and Kurbazh–Nekorkin models, described by discrete maps. A detailed review of these models is presented in [2]. Biological adequacy and the presence of conformity of model and physiological parameters are advantages of many continuous-time models. However, such models are more difficult to study than discrete-time models, especially when simulating large ensembles of connected neurons. Models with continuous time, as a rule, can reproduce the neural generation of either only spikes (individual impulses) or only bursts (groups of two or more successive spikes or spikes alternated by periods of inactivity). Mishchenko et al. [3] proposed a continuous-time spike and burst generation model based on a phaselocked loop [4] devoid of this drawback. When changing the parameters, such a model can generate both single spikes and bursts, including those with different numbers of spikes in neighboring bursts. However, the model [3] cannot generate complex modes, in which switching between different types of neural activity takes place. Chaotic spike-burst oscillations are typical of real neurons, while regimes in that relatively regular fluctuations in the membrane potential of neurons are alternated with irregular ones are characteristic of epilepsy [5].
In the present work, we propose a modification of the neural activity model [3, 6], which consists in introducing the dependence of one of the model dynamic variables on the delay time. The dynamics of the modified model is described by the following system of differential equations with a delayed argument:
dϕ = y, dt
dy = z, dt
(1) dz ε1ε2 = γ − (ε1 + ε2 )z − (1 + ε1 cos ϕ)y(t − τ). dt In terms of a phase-locked loop, variables ϕ and y describe the phase difference and the frequency difference of the tuned and reference oscillators, respect
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